Archimedes' theorem displaces a body immersed in a liquid. Archimedes' Law: the history of discovery and the essence of the phenomenon for dummies
Nick. Gorkavy
Other scientific tales Nick. Gorkavoy were published in the journal “Science and Life” in 2010-2013.
Domenico Fetti. Archimedes is thinking. 1620 Painting from the Old Masters Gallery, Dresden.
Edward Vimon. Death of Archimedes. 1820s.
Tomb of Archimedes in Syracuse. Photo: Codas2.
Ortygia Island, the historical center of Syracuse, the hometown of Archimedes. On these shores, Archimedes burned and sank Roman galleys. Photo: Marcos90.
Greek Theater in Syracuse. Photo: Victoria|photographer_location_London, UK.
Archimedes turns the Earth over with a lever. Ancient engraving. 1824
The image of Archimedes on the Fields Gold Medal, the highest honor given to mathematicians. The inscription in Latin: “Transire suum pectus mundoque potiri” - “To transcend your human limitations and conquer the Universe.” Photo by Stefan Zachov.
Each new tale of the writer and astrophysicist, doctor of physical and mathematical sciences Nikolai Nikolaevich Gorkavy (Nick. Gorkavy) is a story about how important discoveries were made in one or another field of science. And it is no coincidence that the heroes of his popular science novels and fairy tales were Princess Dzintara and her children - Galatea and Andrei, because they are from the breed of those who strive to “know everything”. The stories told by Dzintara to children were included in the collection “Star Vitamin”. It turned out to be so interesting that readers demanded a continuation. We invite you to familiarize yourself with some fairy tales from the future collection “The Makers of Times.” Here is the first publication.
The greatest scientist of the ancient world, the ancient Greek mathematician, physicist and engineer Archimedes (287-212 BC) was from Syracuse - a Greek colony on the largest island of the Mediterranean - Sicily. The ancient Greeks, the creators of European culture, settled there almost three thousand years ago - in the 8th century BC, and by the time of the birth of Archimedes, Syracuse was a thriving cultural city, home to its philosophers and scientists, poets and orators.
The stone houses of the townspeople surrounded the palace of the king of Syracuse, Hieron II, and high walls protected the city from enemies. Residents loved to gather in stadiums, where runners and discus throwers competed, and in bathhouses, where they not only washed, but relaxed and exchanged news.
That day, the baths on the main square of the city were noisy - laughter, screams, splashing water. Young people swam in a large pool, and older people, holding silver goblets of wine in their hands, had a leisurely conversation on comfortable couches. The sun peeked into the courtyard of the baths, illuminating the doorway leading to a separate room. In it, in a small pool that looked like a bathtub, sat alone a man who behaved completely differently from the others. Archimedes - and it was he - closed his eyes, but by some elusive signs it was clear that this man was not sleeping, but was thinking intensely. In recent weeks, the scientist became so deep in his thoughts that he often forgot even about food and his family had to make sure that he did not go hungry.
It began with the fact that King Hieron II invited Archimedes to his palace, poured him the best wine, asked about his health, and then showed him a golden crown made for the ruler by the court jeweler.
“I don’t know much about jewelry, but I do know about people,” Hieron said. - And I think that the jeweler is deceiving me.
The king took a gold bar from the table.
I gave him the exact same ingot and he made a crown out of it. The weight of the crown and the ingot are the same, my servant checked this. But I still have doubts: is there silver mixed into the crown? You, Archimedes, are the greatest scientist of Syracuse, and I ask you to check this, because if the king puts on a false crown, even the street boys will laugh at him...
The ruler handed the crown and ingot to Archimedes with the words:
If you answer my question, you will keep the gold for yourself, but I will still be your debtor.
Archimedes took the crown and the gold bar, left the royal palace, and from then on lost peace and sleep. If he can’t solve this problem, then no one can either. Indeed, Archimedes was the most famous scientist of Syracuse, studied in Alexandria, was friends with the head of the Library of Alexandria, mathematician, astronomer and geographer Eratosthenes and other great thinkers of Greece. Archimedes became famous for his many discoveries in mathematics and geometry, laid the foundations of mechanics, and was responsible for several outstanding inventions.
The puzzled scientist came home, put the crown and the ingot on the scales, lifted them by the middle and made sure that the weight of both objects was the same: the bowls swayed at the same level. Archimedes knew the density of pure gold; he had to find out the density of the crown (weight divided by volume). If there is silver in the crown, its density should be less than that of gold. And since the weights of the crown and the ingot are the same, then the volume of the false crown should be greater than the volume of the gold ingot. The volume of the ingot can be measured, but how can one determine the volume of the crown, which has so many complexly shaped teeth and petals? This problem tormented the scientist. He was an excellent geometer, for example, he solved a difficult problem - determining the area and volume of a sphere and a cylinder circumscribed around it, but how to find the volume of a body of complex shape? A fundamentally new solution is needed.
Archimedes came to the bathhouse to wash off the dust of a hot day and refresh his head, tired from thinking. Ordinary people, while bathing in a bathhouse, could chat and chew figs, but Archimedes’ thoughts about the unsolved problem did not leave him either day or night. His brain searched for a solution, clinging to any clue.
Archimedes took off his chiton, put it on the bench and walked up to the small pool. Water splashed in it three fingers below the edge. When the scientist plunged into the water, its level rose noticeably, and the first wave even splashed onto the marble floor. The scientist closed his eyes, enjoying the pleasant coolness. Thoughts about the volume of the crown habitually swirled in my head.
Suddenly Archimedes felt that something important had happened, but could not understand what. He opened his eyes in annoyance. Voices and someone's heated argument were heard from the direction of the large pool - it seemed about the last law of the ruler of Syracuse. Archimedes froze, trying to understand what had happened? He looked around: the water in the pool did not reach the edge by only one finger, and yet when he entered the water, its level was lower.
Archimedes stood up and left the pool. When the water calmed down, she was again three fingers below the edge. The scientist climbed into the pool again - the water obediently rose. Archimedes quickly estimated the size of the pool, calculated its area, then multiplied it by the change in water level. It turned out that the volume of water displaced by his body is equal to the volume of the body, if we assume that the densities of water and the human body are almost the same and each cubic decimeter, or a cube of water with a side of ten centimeters, can be equated to a kilogram of the weight of the scientist himself. But during the dive, Archimedes’ body lost weight and floated in the water. In some mysterious way, the water displaced by the body took away his weight...
Archimedes realized that he was on the right path, and inspiration carried him on its mighty wings. Is it possible to apply the found law on the volume of displaced fluid to the crown? Certainly! You need to lower the crown into water, measure the increase in the volume of liquid, and then compare it with the volume of water displaced by the gold bar. Problem solved!
According to legend, Archimedes, with a victorious cry of “Eureka!”, which means “Found!” in Greek, jumped out of the pool and, forgetting to put on his chiton, rushed home. I urgently needed to check my decision! He ran through the city, and the residents of Syracuse waved their hands at him in greeting. Still, it’s not every day that the most important law of hydrostatics is discovered, and it’s not every day that you can see a naked man running through the central square of Syracuse.
The next day the king was informed about the arrival of Archimedes.
“I solved the problem,” said the scientist. - There really is a lot of silver in the crown.
How did you know this? - the ruler asked.
Yesterday, in the baths, I guessed that a body that is immersed in a pool of water displaces a volume of liquid equal to the volume of the body itself, and at the same time loses weight. Returning home, I conducted many experiments with scales immersed in water, and proved that a body in water loses exactly as much weight as the liquid it displaces weighs. Therefore, a person can swim, but a gold bar cannot, but it still weighs less in water.
And how does this prove the presence of silver in my crown? - asked the king.
“Tell me to bring a vat of water,” Archimedes asked and took out the scales. While the servants were dragging the vat to the royal chambers, Archimedes put the crown and the ingot on the scales. They balanced each other.
If there is silver in the crown, then the volume of the crown is greater than the volume of the ingot. This means that when immersed in water, the crown will lose more weight and the scales will change their position,” said Archimedes and carefully immersed both scales in the water. The bowl with the crown immediately rose up.
You are truly a great scientist! - exclaimed the king. - Now I can order a new crown for myself and check whether it is real or not.
Archimedes hid a grin in his beard: he understood that the law he had discovered the day before was much more valuable than a thousand golden crowns.
Archimedes' law has remained in history forever; it is used when designing any ships. Hundreds of thousands of ships ply the oceans, seas and rivers, and each of them floats on the surface of the water thanks to the force discovered by Archimedes.
When Archimedes grew old, his measured studies in science suddenly ended, as did the quiet life of the townspeople - the rapidly growing Roman Empire decided to conquer the fertile island of Sicily.
In 212 BC. a huge fleet of galleys filled with Roman soldiers approached the island. The advantage in strength of the Romans was obvious, and the commander of the fleet had no doubt that Syracuse would be captured very quickly. But that was not the case: as soon as the galleys approached the city, powerful catapults struck from the walls. They threw heavy stones so accurately that the invaders' galleys were shattered into splinters.
The Roman commander was not at a loss and commanded the captains of his fleet:
Come to the very walls of the city! At close range, catapults will not be afraid of us, and archers will be able to shoot accurately.
When the fleet, with losses, broke through to the city walls and prepared to storm it, a new surprise awaited the Romans: now light throwing vehicles pelted them with a hail of cannonballs. The lowering hooks of powerful cranes grabbed the Roman galleys by the bows and lifted them into the air. The galleys overturned, fell down and sank.
The famous ancient historian Polybius wrote about the assault on Syracuse: “The Romans could quickly take possession of the city if someone had removed one old man from among the Syracusans.” This old man was Archimedes, who designed throwing machines and powerful cranes to protect the city.
The quick capture of Syracuse failed, and the Roman commander gave the command to retreat. The greatly reduced fleet retreated to a safe distance. The city held firm thanks to the engineering genius of Archimedes and the courage of the townspeople. The scouts reported to the Roman commander the name of the scientist who created such an impregnable defense. The commander decided that after the victory he needed to get Archimedes as the most valuable military trophy, because he alone was worth an entire army!
Day after day, month after month, men stood guard on the walls, shot with bows and loaded catapults with heavy stones, which, alas, did not reach their target. The boys brought water and food to the soldiers, but they were not allowed to fight - they were still too young!
Archimedes was old, he, like children, could not shoot from a bow as far as young and strong men, but he had a powerful brain. Archimedes gathered the boys and asked them, pointing to the enemy galleys:
Want to destroy the Roman fleet?
We are ready, tell us what to do!
The wise old man explained that he would have to work hard. He ordered each boy to take a large copper sheet from the already prepared pile and place it on smooth stone slabs.
Each of you must polish the sheet so that it shines in the sun like gold. And then tomorrow I will show you how to sink Roman galleys. Work, friends! The better you polish the copper today, the easier it will be for us to fight tomorrow.
Will we fight ourselves? - asked the little curly boy.
Yes,” Archimedes said firmly, “tomorrow you will all be on the battlefield along with the soldiers.” Each of you will be able to accomplish a feat, and then legends and songs will be written about you.
It is difficult to describe the enthusiasm that gripped the boys after Archimedes' speech, and they energetically began polishing their copper sheets.
The next day, at noon, the sun burned scorchingly in the sky, and the Roman fleet stood motionless at anchor in the outer roadstead. The wooden sides of the enemy galleys heated up in the sun and oozed resin, which was used to protect the ships from leaks.
Dozens of teenagers gathered on the fortress walls of Syracuse, where enemy arrows could not reach. In front of each of them stood a wooden shield with a polished copper sheet. The shield supports were made so that the copper sheet could be easily turned and tilted.
“Now we’ll check how well you polished the copper,” Archimedes addressed them. - I hope everyone knows how to make sunbeams?
Archimedes approached the little curly-haired boy and said:
Catch the sun with your mirror and direct the sunbeam into the middle of the side of the large black galley, just under the mast.
The boy rushed to carry out the instructions, and the warriors crowded on the walls looked at each other in surprise: what else was the cunning Archimedes up to?
The scientist was pleased with the result - a spot of light appeared on the side of the black galley. Then he turned to the other teenagers:
Point your mirrors at the same place!
Wooden supports creaked, copper sheets rattled - a flock of sunbeams ran towards the black galley, and its side began to fill with bright light. The Romans poured onto the decks of the galleys - what was happening? The commander-in-chief came out and also stared at the sparkling mirrors on the walls of the besieged city. Gods of Olympus, what else did these stubborn Syracusans come up with?
Archimedes instructed his army:
Keep your eyes on the sunbeams - let them always be directed to one place.
Not even a minute had passed before smoke began to billow from a shining spot on board the black galley.
Water, water! - the Romans shouted. Someone rushed to draw sea water, but the smoke quickly gave way to flames. The dry, tarred wood burned beautifully!
Move the mirrors to the adjacent galley on the right! - Archimedes commanded.
In a matter of minutes, the neighboring galley also began to fire. The Roman naval commander came out of his stupor and ordered to weigh anchor in order to move away from the walls of the cursed city with its main defender Archimedes.
Unfastening the anchors, putting the rowers on the oars, turning the huge ships around and taking them out to sea at a safe distance is not a quick task. While the Romans were running hecticly along the decks, choking from the choking smoke, the young Syracusans were transferring mirrors to new ships. In the confusion, the galleys came so close to each other that the fire spread from one ship to another. In their haste to set sail, some ships unfurled their sails, which, as it turned out, burned no worse than the tar sides.
Soon the battle was over. Many Roman ships burned out in the roadstead, and the remnants of the fleet retreated from the city walls. There were no losses among the young army of Archimedes.
Glory to the great Archimedes! - the delighted residents of Syracuse shouted and thanked and hugged their children. A mighty warrior in shining armor firmly shook the curly-haired boy's hand. His small palm was covered with bloody calluses and abrasions from polishing the copper sheet, but he did not even wince when shaking hands.
Well done! - the warrior said respectfully. “The people of Syracuse will remember this day for a long time.”
Two millennia passed, but this day remained in history, and not only the Syracusans remembered it. Residents of different countries know the amazing story of Archimedes burning Roman galleys, but he alone would not have done anything without his young assistants. By the way, quite recently, already in the twentieth century AD, scientists conducted experiments that confirmed the full functionality of the ancient “superweapon” invented by Archimedes to protect Syracuse from invaders. Although there are historians who consider this a legend...
Oh, it's a pity I wasn't there! - exclaimed Galatea, who was listening attentively with her brother to the evening fairy tale that their mother, Princess Dzintara, was telling them. She continued reading the book:
Having lost hope of capturing the city by force of arms, the Roman commander resorted to the old tried and tested method - bribery. He found traitors in the city, and Syracuse fell. The Romans stormed into the city.
Find me Archimedes! - ordered the commander. But the soldiers, intoxicated by victory, did not understand well what he wanted from them. They broke into houses, robbed and killed. One of the warriors ran out to the square where Archimedes was working, drawing a complex geometric figure in the sand. Soldiers' boots trampled the fragile drawing.
Don't touch my drawings! - Archimedes said menacingly.
The Roman did not recognize the scientist and struck him with a sword in anger. This is how this great man died.
Archimedes' fame was so great that his books were often rewritten, thanks to which a number of works have survived to this day, despite the fires and wars of two millennia. The history of the books of Archimedes that have come down to us was often dramatic. It is known that in the 13th century, some ignorant monk took the book of Archimedes, written on durable parchment, and washed away the formulas of the great scientist in order to get blank pages to write down prayers. Centuries passed, and this prayer book fell into the hands of other scientists. Using a strong magnifying glass, they examined its pages and discerned traces of the erased precious text of Archimedes. The book of the brilliant scientist was restored and printed in large quantities. Now it will never disappear.
Archimedes was a real genius who made many discoveries and inventions. He was ahead of his contemporaries not even by centuries - by millennia.
In the book “Psammitus, or Calculus of Grains of Sand,” Archimedes retold the bold theory of Aristarchus of Samos, according to which the great Sun is located in the center of the world. Archimedes wrote: “Aristarchus of Samos... believes that the fixed stars and the Sun do not change their place in space, that the Earth moves in a circle around the Sun, located at its center...” Archimedes considered the heliocentric theory of Samos convincing and used it to estimate the size spheres of fixed stars. The scientist even built a planetarium, or “celestial sphere,” where one could observe the movement of the five planets, the rising of the sun and moon, its phases and eclipses.
The rule of leverage, which Archimedes discovered, became the basis of all mechanics. And although the lever was known before Archimedes, he outlined its complete theory and successfully applied it in practice. In Syracuse, he single-handedly launched the new multi-deck ship of the king of Syracuse, using an ingenious system of blocks and levers. It was then, appreciating the full power of his invention, that Archimedes exclaimed: “Give me a fulcrum, and I will turn the world around.”
Archimedes' achievements in the field of mathematics, which, according to Plutarch, he was simply obsessed with, are invaluable. His main mathematical discoveries relate to mathematical analysis, where the scientist’s ideas formed the basis of integral and differential calculus. The ratio of the circumference of a circle to its diameter, calculated by Archimedes, was of great importance for the development of mathematics. Archimedes gave an approximation for the number π (Archimedean number):
The scientist considered his highest achievement to be his work in the field of geometry and, above all, the calculation of a ball inscribed in a cylinder.
What kind of cylinder and ball? - asked Galatea. - Why was he so proud of them?
Archimedes was able to show that the area and volume of a sphere are related to the area and volume of the described cylinder as 2:3.
Dzintara rose and removed from the shelf a model of the globe, which was soldered inside a transparent cylinder so that it was in contact with it at the poles and at the equator.
I have loved this geometric toy since childhood. Look, the area of the ball is equal to the area of four circles of the same radius or the area of the side of a transparent cylinder. If you add the areas of the base and top of the cylinder, it turns out that the area of the cylinder is one and a half times the area of the ball inside it. The same relationship holds for the volumes of a cylinder and a sphere.
Archimedes was delighted with the result. He knew how to appreciate the beauty of geometric figures and mathematical formulas - that is why it is not a catapult or a burning galley that adorns his grave, but the image of a ball inscribed in a cylinder. Such was the desire of the great scientist.
It would seem that there is nothing simpler than Archimedes' law. But once upon a time Archimedes himself really puzzled over his discovery. How it was?
There is an interesting story connected with the discovery of the fundamental law of hydrostatics.
Interesting facts and legends from the life and death of Archimedes
In addition to such a gigantic breakthrough as the discovery of Archimedes’ law itself, the scientist has a whole list of merits and achievements. In general, he was a genius who worked in the fields of mechanics, astronomy, and mathematics. He wrote such works as a treatise “on floating bodies”, “on the ball and cylinder”, “on spirals”, “on conoids and spheroids” and even “on grains of sand”. The latest work attempted to measure the number of grains of sand needed to fill the Universe.
Role of Archimedes in the Siege of Syracuse
In 212 BC, Syracuse was besieged by the Romans. 75-year-old Archimedes designed powerful catapults and light short-range throwing machines, as well as the so-called “Archimedes claws”. With their help it was possible to literally turn over enemy ships. Faced with such powerful and technological resistance, the Romans were unable to take the city by storm and were forced to begin a siege. According to another legend, Archimedes, using mirrors, managed to set fire to the Roman fleet, focusing the sun's rays on the ships. The veracity of this legend seems doubtful, because None of the historians of that time mentioned this.
Death of Archimedes
According to many testimonies, Archimedes was killed by the Romans when they finally took Syracuse. Here is one of the possible versions of the death of the great engineer.
On the porch of his house, the scientist was thinking about the diagrams that he drew with his hand right in the sand. A passing soldier stepped on the drawing, and Archimedes, deep in thought, shouted: “Get away from my drawings.” In response to this, a soldier hurrying somewhere simply pierced the old man with a sword.
Well, now about the sore point: about the law and power of Archimedes...
How Archimedes' law was discovered and the origin of the famous "Eureka!"
Antiquity. Third century BC. Sicily, where there is still no mafia, but there are ancient Greeks.
An inventor, engineer and theoretical scientist from Syracuse (a Greek colony in Sicily), Archimedes served under King Hiero II. One day, jewelers made a golden crown for the king. The king, being a suspicious person, summoned the scientist to his place and instructed him to find out whether the crown contained silver impurities. Here it must be said that at that distant time no one had resolved such issues and the case was unprecedented.
Archimedes thought for a long time, came up with nothing, and one day decided to go to the bathhouse. There, sitting down in a basin of water, the scientist found a solution to the problem. Archimedes drew attention to a completely obvious thing: a body, immersed in water, displaces a volume of water equal to the body’s own volume.
It was then that, without even bothering to get dressed, Archimedes jumped out of the bathhouse and shouted his famous “Eureka,” which means “found.” Appearing to the king, Archimedes asked to give him ingots of silver and gold, equal in weight to the crown. By measuring and comparing the volume of water displaced by the crown and the ingots, Archimedes discovered that the crown was not made of pure gold, but was mixed with silver. This is the story of the discovery of Archimedes' law.
The essence of Archimedes' law
If you are asking yourself how to understand Archimedes' principle, we will answer. Just sit down, think, and understanding will come. Actually, this law says:
A body immersed in a gas or liquid is subject to a buoyancy force equal to the weight of the liquid (gas) in the volume of the immersed part of the body. This force is called the Archimedes force.
As we can see, the Archimedes force acts not only on bodies immersed in water, but also on bodies in the atmosphere. The force that makes a balloon rise up is the same Archimedes force. The Archimedean force is calculated using the formula:
Here the first term is the density of the liquid (gas), the second is the acceleration of gravity, the third is the volume of the body. If the force of gravity is equal to the force of Archimedes, the body floats, if it is greater, it sinks, and if it is less, it floats until it begins to float.
In this article we looked at Archimedes' law for dummies. If you want to learn how to solve problems where Archimedes' law is found, contact to our specialists. The best authors will be happy to share their knowledge and break down the solution to the most difficult problem “on the shelves.”
One of the first physical laws studied by high school students. Any adult remembers at least approximately this law, no matter how far he is from physics. But sometimes it is useful to return to the exact definitions and formulations - and understand the details of this law that may have been forgotten.
What does Archimedes' law say?
There is a legend that the ancient Greek scientist discovered his famous law while taking a bath. Having plunged into a container filled to the brim with water, Archimedes noticed that the water splashed out - and experienced an epiphany, instantly formulating the essence of the discovery.
Most likely, in reality the situation was different, and the discovery was preceded by long observations. But this is not so important, because in any case, Archimedes managed to discover the following pattern:
- plunging into any liquid, bodies and objects experience several multidirectional forces at once, but directed perpendicular to their surface;
- the final vector of these forces is directed upward, so any object or body, finding itself in a liquid at rest, experiences pushing;
- in this case, the buoyancy force is exactly equal to the coefficient that is obtained if the product of the volume of the object and the density of the liquid is multiplied by the acceleration of free fall.
So, Archimedes established that a body immersed in a liquid displaces a volume of liquid that is equal to the volume of the body itself. If only part of a body is immersed in a liquid, then it will displace the liquid, the volume of which will be equal to the volume of only the part that is immersed.
The same principle applies to gases - only here the volume of the body must be correlated with the density of the gas.
You can formulate a physical law a little more simply - the force that pushes an object out of a liquid or gas is exactly equal to the weight of the liquid or gas displaced by this object during immersion.
The law is written in the form of the following formula:
What is the significance of Archimedes' law?
The pattern discovered by the ancient Greek scientist is simple and completely obvious. But at the same time, its importance for everyday life cannot be overestimated.
It is thanks to the knowledge of the pushing of bodies by liquids and gases that we can build river and sea vessels, as well as airships and balloons for aeronautics. Heavy metal ships do not sink due to the fact that their design takes into account Archimedes' law and numerous consequences from it - they are built so that they can float on the surface of the water, and do not sink. Aeronautics operate on a similar principle - they use the buoyancy of air, becoming, as it were, lighter in the process of flight.
Archimedes' law is the law of statics of liquids and gases, according to which a body immersed in a liquid (or gas) is acted upon by a buoyant force equal to the weight of the liquid in the volume of the body.
Background
"Eureka!" (“Found!”) - this is the exclamation, according to legend, made by the ancient Greek scientist and philosopher Archimedes, who discovered the principle of repression. Legend has it that the Syracusan king Heron II asked the thinker to determine whether his crown was made of pure gold without harming the royal crown itself. It was not difficult to weigh the crown of Archimedes, but this was not enough - it was necessary to determine the volume of the crown in order to calculate the density of the metal from which it was cast and determine whether it was pure gold. Then, according to legend, Archimedes, preoccupied with thoughts about how to determine the volume of the crown, plunged into the bath - and suddenly noticed that the water level in the bath had risen. And then the scientist realized that the volume of his body displaced an equal volume of water, therefore, the crown, if lowered into a basin filled to the brim, would displace a volume of water equal to its volume. A solution to the problem was found and, according to the most common version of the legend, the scientist ran to report his victory to the royal palace, without even bothering to get dressed.
However, what is true is true: it was Archimedes who discovered the principle of buoyancy. If a solid body is immersed in a liquid, it will displace a volume of liquid equal to the volume of the part of the body immersed in the liquid. The pressure that previously acted on the displaced liquid will now act on the solid body that displaced it. And, if the buoyant force acting vertically upward turns out to be greater than the force of gravity pulling the body vertically downward, the body will float; otherwise it will sink (drown). In modern language, a body floats if its average density is less than the density of the liquid in which it is immersed.
Archimedes' Law and Molecular Kinetic Theory
In a fluid at rest, pressure is produced by the impacts of moving molecules. When a certain volume of liquid is displaced by a solid body, the upward impulse of the collisions of molecules will fall not on the liquid molecules displaced by the body, but on the body itself, which explains the pressure exerted on it from below and pushing it towards the surface of the liquid. If the body is completely immersed in the liquid, the buoyant force will continue to act on it, since the pressure increases with increasing depth, and the lower part of the body is subjected to more pressure than the upper, which is where the buoyant force arises. This is the explanation of buoyant force at the molecular level.
This pushing pattern explains why a ship made of steel, which is much denser than water, remains afloat. The fact is that the volume of water displaced by a ship is equal to the volume of steel submerged in water plus the volume of air contained inside the ship's hull below the waterline. If we average the density of the shell of the hull and the air inside it, it turns out that the density of the ship (as a physical body) is less than the density of water, therefore the buoyancy force acting on it as a result of upward impulses of impact of water molecules turns out to be higher than the gravitational force of attraction of the Earth, pulling the ship towards to the bottom - and the ship floats.
Formulation and explanations
The fact that a certain force acts on a body immersed in water is well known to everyone: heavy bodies seem to become lighter - for example, our own body when immersed in a bath. When swimming in a river or sea, you can easily lift and move very heavy stones along the bottom - ones that cannot be lifted on land. At the same time, lightweight bodies resist immersion in water: sinking a ball the size of a small watermelon requires both strength and dexterity; It will most likely not be possible to immerse a ball with a diameter of half a meter. It is intuitively clear that the answer to the question - why a body floats (and another sinks) is closely related to the effect of the liquid on the body immersed in it; one cannot be satisfied with the answer that light bodies float and heavy ones sink: a steel plate, of course, will sink in water, but if you make a box out of it, then it can float; however, her weight did not change.
The existence of hydrostatic pressure results in a buoyant force acting on any body in a liquid or gas. Archimedes was the first to determine the value of this force in liquids experimentally. Archimedes' law is formulated as follows: a body immersed in a liquid or gas is subject to a buoyancy force equal to the weight of the amount of liquid or gas that is displaced by the immersed part of the body.
Formula
The Archimedes force acting on a body immersed in a liquid can be calculated by the formula: F A = ρ f gV Fri,
where ρl is the density of the liquid,
g – free fall acceleration,
Vpt is the volume of the body part immersed in the liquid.
The behavior of a body located in a liquid or gas depends on the relationship between the modules of gravity Ft and the Archimedean force FA, which act on this body. The following three cases are possible:
1) Ft > FA – the body sinks;
2) Ft = FA – the body floats in liquid or gas;
3) Ft< FA – тело всплывает до тех пор, пока не начнет плавать.
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Introduction
Relevance: If you take a close look at the world around you, you can discover many events happening around you. Since ancient times, man has been surrounded by water. When we swim in it, our body pushes some forces to the surface. I have long asked myself the question: “Why do bodies float or sink? Does water push things out?
My research work is aimed at deepening the knowledge gained in class about Archimedean force. Answer the questions that interest me, using life experience, observations of the surrounding reality, conduct my own experiments and explain their results, which will expand my knowledge on this topic. All sciences are interconnected. And the common object of study of all sciences is man “plus” nature. I am sure that the study of the action of Archimedean force is relevant today.
Hypothesis: I assume that at home you can calculate the magnitude of the buoyancy force acting on a body immersed in a liquid and determine whether it depends on the properties of the liquid, the volume and shape of the body.
Object of study: Buoyancy force in liquids.
Tasks:
Study the history of the discovery of Archimedean force;
Study educational literature on the action of Archimedean force;
Develop skills in conducting independent experiments;
Prove that the value of the buoyant force depends on the density of the liquid.
Research methods:
Research;
Calculated;
Information search;
Observations
1. Discovery of the power of Archimedes
There is a famous legend about how Archimedes ran down the street and shouted “Eureka!” This just tells the story of his discovery that the buoyant force of water is equal in magnitude to the weight of the water displaced by it, the volume of which is equal to the volume of the body immersed in it. This discovery is called Archimedes' law.
In the 3rd century BC, there lived Hiero, the king of the ancient Greek city of Syracuse, and he wanted to make himself a new crown from pure gold. I measured it exactly as needed and gave the order to the jeweler. A month later, the master returned the gold in the form of a crown and it weighed as much as the mass of the given gold. But anything can happen, and the master could have cheated by adding silver or, even worse, copper, because you can’t tell the difference by eye, but the mass is what it should be. And the king wants to know: was the work done honestly? And then, he asked the scientist Archimedes to check whether the master made his crown from pure gold. As is known, the mass of a body is equal to the product of the density of the substance from which the body is made and its volume: . If different bodies have the same mass, but they are made of different substances, then they will have different volumes. If the master had returned to the king not a jewelry-made crown, the volume of which is impossible to determine due to its complexity, but a piece of metal of the same shape that the king gave him, then it would have been immediately clear whether he had mixed another metal into it or not. And while taking a bath, Archimedes noticed that water was pouring out of it. He suspected that it was pouring out in exactly the same volume as the volume occupied by his body parts immersed in water. And it dawned on Archimedes that the volume of the crown can be determined by the volume of water displaced by it. Well, if you can measure the volume of the crown, then it can be compared with the volume of a piece of gold of equal mass. Archimedes immersed the crown in water and measured how the volume of water increased. He also immersed a piece of gold in water, the mass of which was the same as that of the crown. And then he measured how the volume of water increased. The volumes of water displaced in the two cases turned out to be different. Thus, the master was exposed as a deceiver, and science was enriched with a remarkable discovery.
It is known from history that the problem of the golden crown prompted Archimedes to study the question of the floating of bodies. The experiments carried out by Archimedes were described in the essay “On Floating Bodies,” which has come down to us. The seventh sentence (theorem) of this work was formulated by Archimedes as follows: bodies heavier than the liquid, immersed in this liquid, will sink until they reach the very bottom, and in the liquid they will become lighter by the weight of the liquid in a volume equal to the volume of the immersed body.
It is interesting that the Archimedes force is zero when a body immersed in a liquid is tightly pressed to the bottom with its entire base.
The discovery of the fundamental law of hydrostatics is the greatest achievement of ancient science.
2. Formulation and explanation of Archimedes' law
Archimedes' law describes the effect of liquids and gases on a body immersed in them, and is one of the main laws of hydrostatics and gas statics.
Archimedes' law is formulated as follows: a body immersed in a liquid (or gas) is acted upon by a buoyant force equal to the weight of the liquid (or gas) in the volume of the immersed part of the body - this force is called by the power of Archimedes:
,
where is the density of the liquid (gas), is the acceleration of gravity, is the volume of the submerged part of the body (or the part of the volume of the body located below the surface).
Consequently, the Archimedean force depends only on the density of the liquid in which the body is immersed and on the volume of this body. But it does not depend, for example, on the density of the substance of a body immersed in a liquid, since this quantity is not included in the resulting formula.
It should be noted that the body must be completely surrounded by liquid (or intersect with the surface of the liquid). So, for example, Archimedes' law cannot be applied to a cube that lies at the bottom of a tank, hermetically touching the bottom.
3. Definition of Archimedes' force
The force with which a body in a liquid is pushed by it can be determined experimentally using this device:
We hang a small bucket and a cylindrical body on a spring fixed to a tripod. We mark the stretch of the spring with an arrow on a tripod, showing the weight of the body in the air. Having lifted the body, we place a glass with a drainage tube under it, filled with liquid to the level of the drainage tube. After which the body is immersed entirely in liquid. In this case, part of the liquid, the volume of which is equal to the volume of the body, is poured from the casting vessel into the glass. The spring pointer rises and the spring contracts, indicating a decrease in body weight in the liquid. In this case, along with the force of gravity, the body is also acted upon by a force that pushes it out of the liquid. If liquid from a glass is poured into the bucket (i.e., the liquid that was displaced by the body), then the spring pointer will return to its initial position.
Based on this experiment, we can conclude that the force pushing out a body completely immersed in a liquid is equal to the weight of the liquid in the volume of this body. The dependence of pressure in a liquid (gas) on the depth of immersion of a body leads to the appearance of a buoyant force (Archimedes' force) acting on any body immersed in a liquid or gas. When a body dives, it moves downward under the influence of gravity. The Archimedean force is always directed opposite to the force of gravity, therefore the weight of a body in a liquid or gas is always less than the weight of this body in a vacuum.
This experiment confirms that the Archimedean force is equal to the weight of the liquid in the volume of the body.
4. Condition of floating bodies
A body located inside a liquid is acted upon by two forces: the force of gravity, directed vertically downward, and the Archimedean force, directed vertically upward. Let us consider what will happen to the body under the influence of these forces if at first it was motionless.
In this case, three cases are possible:
1) If the force of gravity is greater than the Archimedean force, then the body goes down, that is, it sinks:
, then the body drowns;
2) If the modulus of gravity is equal to the modulus of Archimedean force, then the body can be in equilibrium inside the liquid at any depth:
, then the body floats;
3) If the Archimedean force is greater than the force of gravity, then the body will rise from the liquid - float:
, then the body floats.
If a floating body partially protrudes above the surface of the liquid, then the volume of the immersed part of the floating body is such that the weight of the displaced liquid is equal to the weight of the floating body.
Archimedean force is greater than gravity if the density of the liquid is greater than the density of the body immersed in the liquid, if
1) =— a body floats in a liquid or gas, 2) >—the body drowns, 3) < — тело всплывает до тех пор, пока не начнет плавать.
It is these principles of the relationship between gravity and Archimedes’ force that are used in shipping. However, huge river and sea vessels made of steel, the density of which is almost 8 times greater than the density of water, float on the water. This is explained by the fact that only a relatively thin hull of the vessel is made of steel, and most of its volume is occupied by air. The average density of the ship turns out to be significantly less than the density of water; therefore, it not only does not sink, but can also accept a large amount of cargo for transportation. Vessels that navigate rivers, lakes, seas and oceans are built from different materials with different densities. The hull of ships is usually made of steel sheets. All internal fastenings that give ships strength are also made of metals. To build ships, different materials are used, which have both higher and lower density compared to water. The weight of water displaced by the underwater part of the vessel is equal to the weight of the vessel with the cargo in the air or the force of gravity acting on the vessel with the cargo.
For aeronautics, balloons were first used, which were previously filled with heated air, now with hydrogen or helium. In order for the ball to rise into the air, it is necessary that the Archimedean force (buoyancy) acting on the ball be greater than the force of gravity.
5. Conducting the experiment
Investigate the behavior of a raw egg in various types of liquids.
Objective: to prove that the value of the buoyant force depends on the density of the liquid.
I took one raw egg and various kinds of liquids (Appendix 1):
The water is clean;
Water saturated with salt;
Sunflower oil.
First, I lowered the raw egg into clean water - the egg sank - “sank to the bottom” (Appendix 2). Then I added a tablespoon of table salt to a glass of clean water, as a result the egg floats (Appendix 3). And finally, I lowered the egg into a glass with sunflower oil - the egg sank to the bottom (Appendix 4).
Conclusion: in the first case, the density of the egg is greater than the density of water and therefore the egg sank. In the second case, the density of salt water is greater than the density of the egg, so the egg floats in the liquid. In the third case, the density of the egg is also greater than the density of sunflower oil, so the egg sank. Therefore, the greater the density of the liquid, the less the force of gravity.
2. The action of Archimedean force on the human body in water.
Determine the density of the human body experimentally, compare it with the density of fresh and sea water and draw a conclusion about the fundamental ability of a person to swim;
Calculate the weight of a person in the air and the Archimedean force acting on a person in water.
First, I measured my body weight using a scale. Then he measured the volume of the body (without the volume of the head). To do this, I poured enough water into the bath so that when I immersed myself in the water, I was completely submerged (except for my head). Next, using a centimeter tape, I marked the distance from the top edge of the bath to the water level ℓ 1, and then when immersed in water ℓ 2. After that, using a pre-graduated three-liter jar, I began to pour water into the bath from level ℓ 1 to level ℓ 2 - this is how I measured the volume of water I displaced (Appendix 5). I calculated the density using the formula:
The force of gravity acting on a body in the air was calculated using the formula: , where is the acceleration of gravity ≈ 10. The value of the buoyancy force was calculated using the formula described in paragraph 2.
Conclusion: The human body is denser than fresh water, which means it drowns in it. It is easier for a person to swim in the sea than in a river, since the density of sea water is greater, and therefore the buoyant force is greater.
Conclusion
In the process of working on this topic, we learned a lot of new and interesting things. The range of our knowledge has increased not only in the field of action of Archimedes’ power, but also in its application in life. Before starting work, we had a far from detailed idea about it. During the experiments, we experimentally confirmed the validity of Archimedes' law and found out that the buoyancy force depends on the volume of the body and the density of the liquid; the higher the density of the liquid, the greater the Archimedean force. The resulting force, which determines the behavior of a body in a liquid, depends on the mass, volume of the body and the density of the liquid.
In addition to the experiments performed, additional literature was studied about the discovery of Archimedes' force, about the floating of bodies, and aeronautics.
Each of you can make amazing discoveries, and for this you do not need to have any special knowledge or powerful equipment. We just need to look a little more carefully at the world around us, be a little more independent in our judgments, and discoveries will not keep you waiting. The reluctance of most people to explore the world around them leaves a lot of scope for the curious in the most unexpected places.
Bibliography
1. Big book of experiments for schoolchildren - M.: Rosman, 2009. - 264 p.
2. Wikipedia: https://ru.wikipedia.org/wiki/Archimedes_Law.
3. Perelman Ya.I. Entertaining physics. - book 1. - Ekaterinburg.: Thesis, 1994.
4. Perelman Ya.I. Entertaining physics. - book 2. - Ekaterinburg.: Thesis, 1994.
5. Peryshkin A.V. Physics: 7th grade: textbook for educational institutions / A.V. Peryshkin. - 16th ed., stereotype. - M.: Bustard, 2013. - 192 p.: ill.
Annex 1
Appendix 2
Appendix 3
Appendix 4
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